Most car dealers will take a credit card. Your plan can work, but you need to stay on top of it! One late payment can make the whole thing backfire. Credit card companies love to pile on interest charges and late fees. Also make sure you’ve got things covered in case one of the balance transfers doesn’t go through, or takes longer than you anticipated.
I know myself well enough to know that this would not work for me. At least once every year or two, I tend to forget about one payment or another and end up paying something late. I’m better off paying cash, or finding the lowest interest rate I can. Credit unions usually have pretty good auto loan rates.